Let us now address the issue of 
gravitational microlensing. 
This is a lensing of stars by other 
stars. 
As you may recall, one possibility for 
the constituentce of baryonic dark matter 
or maybe all of the dark matter was the 
in the form of some under luminous 
substellar objects, which could be brown 
dwarfs or planets or even little black 
holes or neutron stars, 
and gravitational microlensing offers a 
way to test that. 
So if any such object passes along the 
line of sight of some background star, it 
will cause gravitational magnification 
and there, there will be first horizon 
brightness as it approaches the critical 
radius and [UNKNOWN] so it will be a 
symmetric light curve, 
it will also be same at all wavelengths 
because gravitational lensing is 
achromatic. 
How often will this happen depends 
obviously on a cross section and cross 
sections for stars that are very small, 
they're down, because they're so very 
small compared to interstellar distances. 
So in principle for any given star, it'll 
be extremely rare, and therefore, you'd 
have to monitor a large number of stars 
in order to be able to catch one of 
these. 
Here's some of the model light curves for 
microlensing event. 
The peak magnification will depend on how 
well aligned the passage is. 
The, the closer the better, of course. 
And, how long does the event last will 
depend on the velocity of the lens 
relative to the line of sight. 
Faster moving ones will cause shorter 
events, obviously. 
In order to make this practical, one had 
to monitor many millions of stars, 
and one good solution was to look towards 
the large Magellanic Cloud, 
which is about 50 kiloparsecs away, where 
many, many stars can be in the same field 
of view and then monitor for a whil 
looking for such events. 
This was done by a team called MACHO, for 
Massive Compact Halo Object, and also 
another team called OGLE, and since then, 
many others. 
So there is a probability of seeing 
lensing happen to any given star is about 
one part in 10 million per year, then you 
better look at tens of millions of stars. 
Another possible direction is towards the 
galactic bulge where a large number of 
galactic stars will be the same. 
So MACHO and OGLE were just two 
experiments that started the whole thing. 
MACHO monitored almost 12 millions stars 
for several years and OGLE continues to 
monitor about 33 million stars. 
Many others have been started since then, 
and now, this is a really 
well-established field. 
Shown here on the right, is the first 
MACHO event detected, detected in 
microlensing, 
and the two panels show light curving to 
different filters. 
This was an important discriminator 
against variable stars, because variable 
stars will tend to vary differently in 
different filters, whereas gravitational 
lensing is unique in the sense that it 
will work exactly the same for all 
wavelengths. 
And on the left, we see many more 
different events, 
to date there have been probably 
thousands and thousands of microlensing 
discovered by different groups. 
Recall that Einstein radius essentially 
defines the cross section for strong 
lensing. 
And in this case, it would be as a 
critical impact parameter for lens 
passing along the line or close to the 
line of sight. 
Because mass of the lens and there is 
under square root in Einstein radius in 
the area of the cross section is a square 
of that, it will be directly proportional 
to the mass of the lens, 
thus, we can have an insight into the 
masses of the lenses, 
and if we look at the whole population of 
stars, then we can just add them up. 
So the net total fraction of all stars 
that are being lensed per unit time is 
giving you effectively an optical depth 
due to gravitational microlensing. 
The velocities of lenses are indicative 
of the velocity dispersion of their 
populations, say halo, but they do not by 
themselves say anything about lenses, 
and also, the implification says exactly 
nothing about any given lens. 
Now, since their distances involved in 
the expression of the Einstein radius, 
we have to know where the lenses are, and 
so, the interpretation of the result 
depends critically on where do you assume 
the lensing is happening? Are there stars 
near us? Are there stars in a giant cloud 
or somewhere in between, like in a 
galactic halo? 
If you know or assume, somehow, 
velocities of the lenses, 
say we know that the velocity of 
dispersion of halo stars is couple 
hundred kilometers per second, then you 
can infer masses of lenses from the 
durations of the events, 
and typically, those are measured in days 
or tens of days. 
The shorter the event, the smaller the 
mass, so for an, an individual event 
there are only two things to measure. 
We measure duration and the amplitude. 
The amplitude is simply telling you how 
well aligned you are, it doesn't say 
anything about lenses. 
The duration tells you about either their 
velocity dispersion or the individual 
masses, 
and the total number of lenses you see is 
telling you something about their 
density. 
So here are the original MACHO results, 
they're expressed in probability contours 
as a function of the lens mass and a 
fraction of the mass of the galactic halo 
that could be attributed to them, 
assuming they really are in the galactic 
halo. 
The surprising result here was the 
typical mass was supposed to be about 
half solar mass. 
This was unexpected because the only 
thing that could be like that would be 
white dwarfs, which are evolutionary 
remnants of more massive stars, 
but that means that there must have been 
some huge population of projectors of 
those with other consequences which are 
not seen. 
And so most likely the solution of this 
is that lenses were actually not in 
galactic halo. 
They were either in our disc of our 
galaxy or imaginary clouds themselves. 
But even if they were in a galactic halo, 
the sheer frequency of them eliminated 
the possibility that all of the dark 
matter is in form of MACHOs, regardless 
of what matters are brown dwarfs or black 
holes or anything else. 
This was still a very powerful result. 
Sometimes you can't find where the 
lenses. One it sufficiently close by that 
you can use parallax and the other one is 
there is a binary, because then you can 
use geometry of the binary star to infer 
how far it was. 
A more recent twist on this is that this 
kind of measurement is used to discover 
planets around other stars. 
A complimentery method to those, say 
like, say Kepler Satellite that uses 
eclipses or kinematical measurements with 
radial velocities. 
Next time, we will talk about dark 
energy, 
probably the single most understanding 
problem of physical cosmology today.